Li and Xia Boundary Value Problems (2018) 2018:66 https://doi.org/10.1186/s13661-018-0984-0

RESEARCH

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Existence of multiple solutions for a quasilinear Neumann problem with critical exponent Yuanxiao Li*

and Suxia Xia

*

Correspondence: [email protected] College of Science, Henan University of Technology, Zhengzhou, P.R. China

Abstract The main purpose of this paper is to establish the existence and multiplicity of nontrivial solutions for a quasilinear Neumann problem with critical exponent. It is shown, by the methods of the Lions concentration-compactness principle and the mountain pass lemma, that under certain conditions, the existence of nontrivial solutions are obtained. MSC: 35B33; 35J50; 35J62; 35J92 Keywords: Quasilinear elliptic equation; Nontrivial solution; Neumann boundary condition; Critical Sobolev exponent; Lions concentration-compactness principle

1 Introduction In this paper, we consider the following quasilinear elliptic problem with critical Sobolev exponent: ⎧ ⎨–εp  u + V (x)|u|p–2 u = Q(x)|u|p∗ –2 u + P(x)|u|q–2 u, p ⎩|∇u|p–2 ∂u = 0, ∂ν

x ∈ , x ∈ ∂,

(1.1)

where  ⊂ RN is a bounded domain with smooth boundary, p u = div(|∇u|p–2 ∇u), ε > 0, Np , ν denotes the unit outward normal vector with respect to ∂. 1 < p < N , p < q < p∗ = N–p The weight functions V (x), Q(x) and P(x) are continuous on . Such problems arise in the theory of quasiregular and quasiconformal mapping or in the study of non-Newtonian fluids. In the latter case, the p is a characteristic of the medium. Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids. The early study of Laplacian elliptic equation with critical Sobolev exponent was Pohozaev [1], the author established the nonexistence of nontrivial solution to the Dirichlet problems when  is a star-shaped domain with respect to the origin. Later, Brézis and Nirenberg [2] showed the existence of positive solutions by introducing the low-order perturbation terms, and Struwe [3] also obtained the global compactness result. Since then, the study of these elliptic problems with critical growth terms have been paid wide attentions in recent years (see [4–7]). Set p = 2, ε = 1, P(x) = 0, V (x) = λ, then Problem (1.1) © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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reduces to the following semilinear elliptic problem: ⎧ ⎨–u + λu = Q(x)|u|2∗ –2 u,

x ∈ ,

⎩ ∂u ∂v

x ∈ .

= 0,

(1.2)

Comte and Knaap [8] proved that there exists a nontrivial solution of problem (1.2) by variational method if Q(x) = 1 and λ = –μ. Chabrowski and Willem [9] studied this problem with the assumption that the function Q(x) is nonnegative and Hölder continuous, they obtained the existence of least energy solutions by solving minimization problem corresponding to  Sλ =



inf

∗

u∈H 1 (),  Q(x)|u|2 dx=0

(

2  (|∇u|



+ λu2 ) dx

2∗  Q(x)|u|

2

dx) 2∗

.

Subsequently, Chabrowski and Girão [10] investigated the existence and nonexistence of least energy solutions when the function Q(x) has some symmetry properties. For more relevant information as regards the corresponding problems, the interested reader may refer to [11–21] and the references therein. As for quasilinear elliptic problems with critical Sobolev exponent, the existence and multiplicity of solutions have also been studied extensively. Abreu et al. [22] studied the following nonhomogeneous Neumann boundary problems: ⎧ p–1 q ⎪ ⎪ ⎨–p u + λu = u , x ∈ , ⎪ ⎪ ⎩

u > 0,

x ∈ ,

|∇u|p–2 ∂u = ϕ, ∂v

x ∈ ∂,

(1.3)

where p – 1 < q ≤ p∗ – 1, ϕ ∈ C α (), 0 < α < 1, ϕ ≡ 0. They proved that there exists a λ∗ > 0 such that problem (1.3) has at least two positive solutions if λ > λ∗ , has at least one positive solution if λ = λ∗ and has no positive solution if λ < λ∗ relying on the lower and upper solutions method and variational approach. Zhao et al. [23] discussed the quasilinear elliptic problem of the form ⎧ ⎨– u + λ(x)|u|p–2 u = |u|p∗ –2 u + |u|r–2 u, x ∈ , p ⎩|∇u|p–2 ∂u = η|u|p–2 u, x ∈ ∂, ∂v

(1.4)

they showed that there exists at least a nontrivial solution when p < r < p∗ and there exist infinitely many solutions when 1 < r < p by using the Mountain pass theorem and the concentration-compactness principle. Some authors also studied the critical Sobolev exponent for quasilinear equations and the corresponding evolution problems with Neumann boundary conditions, the reader may also refer to [24–37]. Motivated by the results of the above papers, we discuss the existence of nontrivial nonnegative solutions to Problem (1.1) by a variational method. The special features of this problem are the following. Firstly, due to the lack of compactness of the embedding of ∗ W 1,p () → Lp (), we cannot use the standard variational argument directly. In order to overcome this difficulty and obtain the existence of solutions, we have to add restrictions

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on the weight functions Q(x) and P(x) to prove the corresponding functional of Problem (1.1) satisfies (PS)c -condition in a suitable range by the Lions concentration-compactness principle. Secondly, the weight function V (x) may be unbounded near the boundary ∂, which leads to the space W 1,p () is not suitable for our problem. To solve such problem, we have to introduce a suitable weighted Sobolev space. For the sake of convenience, we introducing a new parameter λ = ε–p , then Problem (1.1) may be rewritten as the following problem: ⎧ ⎨– u + λV (x)|u|p–2 u = λQ(x)|u|p∗ –2 u + λP(x)|u|q–2 u, x ∈ , p ⎩|∇u|p–2 ∂u = 0, x ∈ ∂. ∂ν

(1.5)

Throughout this paper, we make some assumptions on the weight functions Q(x), P(x), V (x) as the following: (A1) Q(x), P(x) are continuous on , and Q(x) > 0, P(x) ≥ 0 for x ∈ ; (A2) V (x) is continuous in , and V (x) ≥ 0, V (x) ≡ 0 for x ∈ . Set Qm = maxx∈∂ Q(x), QM = maxx∈ Q(x), PM = maxx∈ P(x). The main results of this paper are the following. Theorem 1.1 Suppose that (A1), (A2) hold, H(0) > 0 and Qm = Q(0). If functions Q(x), V (x) satisfy p N (A3) QM ≤ 2 N–p Qm , and |Q(x) – Q(0)| = o(|x|α ) as x → 0, where 1 < α < p–1 ;  N(p–1) 1 1 r (A4) ∩B(0,δ) V dx < ∞, where r + r = 1, 1 < r < Np+2p–N–p2 –1 , δ > 0. Then Problem (1.5) has at least one nontrivial solution for every λ > 0 and N ≥ 2p, where H(0) will be later determined. p

Theorem 1.2 Suppose that (A1), (A2) hold. If QM > 2 N–p Qm and functions P(x), V (x) satisfy (A5) P(x) ≡ 0 for x ∈ , and V ∈ L1 (). Then there exists a λ∗ > 0 such that Problem (1.5) has at least one nontrivial solution for 0 < λ < λ∗ . p

Theorem 1.3 Suppose that (A1), (A2) hold. If QM > 2 N–p Qm and functions P(x), V (x) satisfy (A6) P(x) > 0 for x ∈ ; (A7) there exist x0 ∈  and constant δ > 0 such that V (x) = 0 for x ∈ B(x0 , δ) ⊂ . Then there exists a λ∗ > 0 such that Problem (1.5) has at least one nontrivial solution for λ > λ∗ . p

Theorem 1.4 Suppose that (A1), (A2) hold. If QM > 2 N–p Qm and functions P(x), V (x) satisfy the conditions (A6) and (A7). Then, for every integer n, there exists a constant n > 0 such that Problem (1.5) has at least n pairs of nontrivial solutions for λ > n .

2 Preliminaries Firstly, we define the weighted Sobolev space   1,p Wλ,V () = u; Di u ∈ Lp (), i = 1, 2, . . . , N, V (x)|u|p dx < +∞ 

Li and Xia Boundary Value Problems (2018) 2018:66

with norm uλ,V = (

Page 4 of 17



1

+ λV (x)|u|p ) dx) p . Obviously, norms uλ,V and uV are  1 1,p equivalent, Wλ,V () → W (), where uV = (  (|∇u|p + V (x)|u|p ) dx) p . By using the Sobolev embedding theorem, we know that there exists a constant Cq > 0 such that p  (|∇u| 1,p

1,p

|u|q ≤ Cq uV ≤ Cq uλV ,

for λ ≥ 1, u ∈ Wλ,V (),

(2.1)

and 1

1,p

|u|q ≤ Cq uV ≤ λ– p Cq uλV ,

for 0 < λ < 1, u ∈ Wλ,V (),

(2.2)

 1 where |u|q = (  |u|q dx) q , q ∈ (p, p∗ ). Next, we give the definition of weak solution to Problem (1.5). 1,p

Definition 2.1 A function u ∈ Wλ,V () is said to be a weak solution of Problem (1.5) if it satisfies 

 |∇u|

p–2



∇u∇ψ dx + λ

V (x)|u|p–2 uψ dx 



∗ –2

Q(x)|u|p

=λ





1,p

∀ψ ∈ Wλ,V ().

P(x)|u|q–2 uψ dx,

uψ dx + λ 

1,p

Thus, the corresponding energy functional of Problem (1.5) is defined in Wλ,V () by Jλ (u) =

1 p



 λ |∇u|p + λV (x)|u|p dx – ∗ p 



∗

Q(x)|u|p dx – 

λ q

 P(x)|u|q dx. 

Let S be the best Sobolev constants, namely  S=

inf

D1,p (RN )\{0}

(



p  |∇u| dx p

p∗ p∗  |u| dx)

,

(2.3)

∗

where D1,p (RN ) = {u ∈ Lp (RN ) : |∇u| ∈ Lp (RN )}. This constant S is achieved by the functional uε given by uε (x) = CNp ε

N–p p2

p  p–N

ε + |x| p–1 p ,

∗

where the constant CNp is chosen such that –p uε = |uε |p –1 in RN (see [22] for details). In order to obtain the existence of solutions to Problem (1.5), we need the following lemma. Lemma 2.1 For each λ > 0, (i) there exist constants βλ , ρλ > 0 such that Jλ (u) ≥ βλ for uλV = ρλ ; 1,p (ii) there exists an u0 ∈ Wλ,V () with u0 ≡ 0 such that Jλ (u0 ) < 0 for u0 λV > ρλ .

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p

p

p

Proof (i) Firstly, we consider the case λ ≥ 1. let uV = ρ p , then ρλ = uλV ≤ λρ p . Using (2.1) and (2.3), we have p∗ 1 λ p Jλ (u) ≥ uλV – ∗ QM S– p p p

≥

1 λ p uλV – ∗ QM S p p

≥

p ρλ



p∗ – p

 |∇u| dx p

pp∗



λ q – PM Cqq uλV q

λ p∗ q uλV – PM Cqq uλV q

p∗ q

p∗ λp 1 λp – p p∗ –p q q–p – ∗ QM S ρ PM Cq ρ – . p p q

Since p < q < p∗ , taking ρ > 0 small enough, there exists a βλ > 0 such that Jλ (u) ≥ βλ for uλV = ρλ . 1 p If 0 < λ < 1, let uV = ρ p , then ρ > ρλ > λ p ρ. Combining (2.2) with (2.3), we see that p∗ 1 λ λ p p∗ q uλV – ∗ QM S– p uV – PM Cqq uV p p q

p∗ 1 1 – p p∗ –p p 1 q q–p > λρ – PM Cq ρ . – QM S ρ p p∗ q

Jλ (u) ≥

Since p < q < p∗ , taking ρ > 0 small enough, there exists a βλ > 0 such that Jλ (u) ≥ βλ for uλV = ρλ . 1,p (ii) For u ∈ Wλ,V () and u ≡ 0, we define 

∗

Jλ (tu) =

tp tp p uλV – ∗ λ p p

∗

Q(x)|u|p dx – 

tq λ q

 P(x)|u|q dx,

t > 0,



it follows from limt→+∞ Jλ (tu) = –∞ that there exists a t0 > 0 such that t0 uλV > ρλ and Jλ (t0 u) < 0. Letting u0 = t0 u, then condition (ii) holds. The proof of Lemma 2.1 is completed.  Define

 c = inf sup Jλ h(t) , h∈ t∈[0,1]

1,p

where  = {h ∈ C([0, 1], Wλ,V ()) | h(0) = 0, h(1) = t0 u = u0 }. Using Lemma 2.1, we know that the energy functional Jλ (u) satisfies the geometry of the mountain pass lemma, then 1,p there exists a (PS)c -sequence {un } ⊂ Wλ,V () such that Jλ (un ) → c, Jλ (un ) → 0 as n → ∞. Lemma 2.2 Assume (A1), (A2) hold, and {un } be a (PS)c -sequence at the level of c for Jλ with c < c∗ = min{

N

Sp

N–p N–p p Nλ p QM

N

,

Sp

N–p N–p p 2Nλ p Qm

1,p

}, then {un } is relatively compact in Wλ,V ().

Proof Firstly, we prove that {un } is bounded. Since Jλ (un ) → c, Jλ (un ) → 0 as n → ∞, we have     λ 1

λ ∗ |∇un |p + λV (x)|un |p dx – ∗ Q(x)|un |p dx – P(x)|un |q dx Jλ (un ) = p  p  q  = c + o(1)un ,

Li and Xia Boundary Value Problems (2018) 2018:66





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p∗

|∇un | + λV (x)|un | dx – λ p

p





Q(x)|un | dx – λ 

P(x)|un |q dx 

= o(1)un . Combining (A1) and (A2), one has c + o(1)un  = ≥

1 1 – p q







  1 1 ∗ |∇un |p + λV (x)|un |p dx + λ – ∗ Q(x)|un |p dx q p 

1 1 p – un λV . p q

1,p

Thus, we can find that {un } is bounded in Wλ,V (). 1,p Next, we prove that {un } is relatively compact in Wλ,V (). Since {un } is bounded in 1,p 1,p Wλ,V (), there exists a subsequence, still denoted by {un } and u ∈ Wλ,V () such that 1,p

un  u

weakly in Wλ,V (),

un  u

weakly in Lp (),

un → u

strongly in Lq (), p ≤ q < p∗ ,

un → u

a.e. in .

∗

By the Lions concentration-compactness principle [38], there exists at most set J, a set of different points {xj }j∈J ⊂ , sets of nonnegative real numbers {μj }j∈J , {νj }j∈J such that |∇un |p  dμ ≥ |∇u|p +



μj δxj ,

j∈J

∗

∗

|un |p  dν = |u|p +



(2.4) νj δxj ,

j∈J

where δx is the Dirac mass at x, and the constants μj , νj satisfying p p∗

≤ μj ,

Sνj S 2

p N

p p∗

νj

≤ μj ,

where xj ∈ ,

(2.5)

where xj ∈ ∂.

(2.6)

Next, we prove μj = 0 and νj = 0, where j ∈ J. In fact, choosing ε > 0 sufficiently small j such that Bε (xi ) ∩ Bε (xj ) = ∅ for i = j, i, j ∈ J. Let φε (x) be a smooth cut off function centered at xj such that 0 ≤ φεj (x) ≤ 1

for |x – xj | < ε,

⎧ ⎨1, |x – x | ≤ ε , j 2 φεj (x) = ⎩0, |x – xj | ≥ ε,

Noting that   Jλ (un ), un φεj (x)   = |∇un |p φεj (x) dx + |∇un |p–2 ∇un ∇φεj (x)un dx 



and

 j 4 ∇φ  ≤ . ε ε

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∗

V (x)|un |p φεj (x) dx – λ

+λ 

Q(x)|un |p φεj (x) dx 



P(x)|un |q φεj (x) dx,

–λ 

and by (2.4), we have 

 |∇un |p φεj (x) dx ≥

lim lim

ε→0 n→∞ 

|∇u|p φεj (x) dx +

lim

ε→0





   j∈J

 μj δxj φεj (x) dx

≥ μj ,

|∇un |p–2 ∇un ∇φεj (x)un dx = 0,

lim lim

ε→0 n→∞ 



V (x)|un |p φεj (x) dx = 0,

lim lim

ε→0 n→∞ 



∗

Q(x)|un |p φεj (x) dx = Q(xj )νj ,

lim lim

ε→0 n→∞ 



P(x)|un |q φεj (x) dx = 0.

lim lim

ε→0 n→∞ 

Thus,   0 = lim lim Jλ (un ), un φεj (x) ≥ μj – λQ(xj )νj . ε→0 n→∞

If νj = 0, by (2.5) and (2.6), we find that N

νj ≥

Sp N

xj ∈ ,

,

N

λ p Q p (xj ) N

νj ≥

Sp N

N

xj ∈ ∂.

,

2λ p Q p (xj ) On the other hand,

 1 c = lim Jλ (un ) – Jλ (un ), un n→∞ p





 1 1 1 1 1 1 ∗ Q(xj )νj = – ∗ λ Q(x)|u|p dx + – λ P(x)|u|q dx + – ∗ λ p p p q p p   j∈J ≥

1  λ Q(xj )νj , N j∈J

consequently, 1 c ≥ λQ(xj )νj ≥ N c≥

1 λQ(xj )νj ≥ N

N

Sp Nλ

N–p p

N–p p

xj ∈ ,

,

QM N

Sp 2Nλ

N–p p

N–p p

,

xj ∈ ∂,

Qm

∗

which is a contradiction. Hence, μj = 0, νj = 0 and we find that un → u strongly in Lp ().

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1,p

Now, we prove that un → u strongly in Wλ,V (). We have   Jλ (un ) – Jλ (u), un – u p



p

= un λ,V + uλ,V – 



–



 |∇un |p–2 ∇un ∇u + λV (x)|un |p–2 un u dx



 |∇u|p–2 ∇u∇un + λV (x)|u|p–2 uun dx – I – II,



where 

 ∗ ∗ Q(x) |un |p –2 un – |u|p –2 u (un – u) dx,

I =λ 



 P(x) |un |q–2 un – |u|q–2 u (un – u) dx.

II = λ 

By the Hölder inequality and Jensen’s inequality (a + b)α (c + d)1–α ≥ aα c1–α + bα d1–α , where α ∈ (0, 1), a > 0, b > 0, c > 0, d > 0, we have 

 |∇un |p–2 ∇un ∇u + λV (x)|un |p–2 un u dx 

 ≤

|∇un | dx p



p–1 p



|∇u| dx p

p1



 

p–1

p1 p p p λ V (x)|u| dx + λ V (x)|un | dx 



 ≤

|∇un |p + λV (x)|un |p dx



p–1 p



|∇u|p + λV (x)|u|p dx



Similarly, we get  



 p–1 |∇u|p–2 ∇u∇un + λV (x)|u|p–2 uun dx ≤ uλV un λV , 

∗ –1

|I| ≤ λQM

|un |p



|u|p



∗

|un |p dx

p∗ –1 p∗





∗

|u|p dx

p∗ –1 p∗

 

|un |

q–1

∗

|un – u|p dx



, 

|u|

q–1

|un – u| dx



 |un | dx q



q–1 q



|un – u| dx q

q1



 |u|q dx

+ λPM 



q–1 q 

1 p∗



|un – u| dx +



≤ λPM

1 p∗



+ λQM |II| ≤ λPM

∗

|un – u|p dx





 |un – u| dx



 ≤ λQM

∗ –1

|un – u| dx +

|un – u|q dx

q1 .

p1

p–1

= un λV uλV .

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We have 

  p–1 p–1 

0 = lim Jλ (un ) – Jλ (u), un – u ≥ lim un λV – uλV un λV – uλV ≥ 0. n→∞

n→∞

1,p

Hence, un → u strongly in Wλ,V ().



Since 0 ∈ ∂ and ∂ ∈ C 2 , the boundary ∂ near the origin can be represented xN =  2 2 h(x ) = 12 N–1 i=1 λi xi + o(|x | ), where x = (x1 , x2 , . . . , xN–1 ) ∈ D(0, δ) = B(0, δ) ∩ {xN = 0}, λi (i = 1, 2, . . . , N – 1) are the principal curvatures of ∂ at 0 and the mean curvatures H(0) = 1 N–1 i=1 λi > 0. Then the following lemma holds. N–1 Lemma 2.3 ([22]) (1) For N > 2p – 1 and ε > 0 small enough,  

p–1  |∇uε |p dx = |∇uε |p dx – K1 (ε) + o ε p , 





∗

|uε |p dx = 

RN +

RN +

p–1  ∗ |uε |p dx – K2 (ε) + o ε p ,

where K1 (ε), K2 (ε) satisfy

 p  3p–2

  p–1 N –p p 1 –N   p–1 p 1 + x  x K1 (ε) = H(0)CNp dx = K1 , 2 p–1 RN–1  p  p–1

  p–1 1 –N  2 p∗ x dx = K2 . lim ε– p K2 (ε) = H(0)CNp 1 + x  ε→0 2 RN–1 p–1

lim ε– p ε→0

(2) ⎧ N–p p ⎪ ⎪ ⎨O(ε N–p ), |uε |p dx = O(ε p | ln ε|), ⎪  ⎪ ⎩ O(εp–1 ),



N < p2 , N = p2 , N > p2 .

(3) ⎧ q(N–p) ⎪ ⎪ p2 ), O(ε q < N(p–1) , ⎪ ⎪ N–p  ⎨ q(N–p) q N(p–1) 2 |uε | dx = O(ε p | ln ε|), q = N–p , ⎪  ⎪ (p–1)(Np–q(N–p)) ⎪ ⎪ ⎩ p2 ), q > N(p–1) . O(ε N–p

3 Proof of main results Let ϕ(x) ∈ C0∞ (RN ) be a smooth cut off function such that δ ≤ |x| ≤ δ; 2 δ |x| < ; 2

0 ≤ ϕ(x) ≤ 1, ϕ(x) = 1, ϕ(x) = 0,

|x| > δ.

Define ωε = ϕuε , then we have the following lemma about ωε .

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Lemma 3.1 Suppose N ≥ 2p, 0 ∈ ∂. If the function V (x) satisfies then  N(p–1) 

N–p V ωεp dx = O ε p +p–N+ pr ,



∩B(0,δ) V

r

dx < ∞,

∩B(0,δ) 1 r

where

+

1 r

= 1, 1 < r <

N(p–1) . Np+2p–N–p2 –1

Proof According to the Hölder inequality and the definition of ωε , we have 

 V ωεp dx

r

≤ ∩B(0,δ)

≤ε

=ε

(N–p) p

 p CNp

r

B(0,δε N(p–1) (N–p)p

1r dx

V dx

1 

p–1 – p )

1r

B(0,δ)





p

CNp

p r(p–N)

ε + |x| p–1 dx

r

∩B(0,δ)

N–p N(p–1) p +p–N+ pr

×

ωεpr

∩B(0,δ)





r

V dx

∩B(0,δ)

Noting that

1 

V r dx

1 r

∩B(0,δ)

p r(p–N)

1 + |x| p–1 dx

1r .

≤ 1 < r, a series of computations yield

N(p–1) 

N–p V ωεp dx = O ε p +p–N+ pr .

∩B(0,δ)



Lemma 3.2 Suppose that (A1), (A2) hold and 0 ∈ ∂, H(0) > 0, Qm = Q(0). If the functions Q(x), V (x) satisfy the conditions (A3), (A4), then there exists a nonnegative function v ∈ 1,p Wλ,V (), v ≡ 0, such that sup Jλ (tv) < c∗

(3.1)

t≥0

for each λ > 0, N ≥ 2p. Proof We divide the proof into three steps. (i) We consider the functional g(t) = Jλ (tωε )    t p∗ tp

∗ |∇ωε |p + λV (x)|ωε |p dx – ∗ λ Q(x)|ωε |p dx = p  p  q  t – λ P(x)|ωε |q dx, t > 0. q  Noting that limt→∞ g(t) = –∞, g(0) = 0, g(t) > 0 for t → 0+ , we know that there exists a tε > 0 such that supt>0 g(t) is attained for tε and tε is uniformly bounded for ε > 0 sufficiently small. Thus, g(tε ) = sup Jλ (tωε ) t≥0

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   t p∗ tp p p∗ ≤ sup |∇ωε | dx – ∗ λ Q(x)|ωε | dx p t≥0 p   p  q  tε tε λV (x)|ωε |p dx – λ P(x)|ωε |q dx + p  q     Np p  p 1 tε  |∇ωε | dx = + λV (x)|ωε |p dx N (λ  Q(x)|ωε |p∗ dx) N–p p  N  q  tε – λ P(x)|ωε |q dx. q  

(3.2)

(ii) When ε > 0 is sufficiently small, we have 



p∗

p–1  ∗ |uε |p dx + o ε p ,

Q(x)|ωε | dx = Qm 







p–1  |∇uε |p dx + o ε p ,

|∇ωε |p dx ≤ 











(3.3)

p–1  |uε |q dx + o ε p ,

|ωε |q dx = 



∗

p–1  ∗ |uε |p dx + o ε p .

|ωε |p dx = 



We firstly prove the first formula. Since |Q(x) – Q(0)| = o(|x|α ) for x → 0, there exists a 0 < δ0 ≤ δ such that |Q(x) – Q(0)| ≤ C|x|α for |x| < δ0 , where C > 0 is constant. Moreover 

  Q(x) – Q(0)|ωε |p∗ dx





  Q(x) – Q(0)|ωε |p∗ dx +

≤ ∩|x|≤δ0



≤C



p∗

|x|≤δ0 p∗

≤ CCNp ε

 ∩|x|≥δ0

|x|α |ωε | dx + 2QM

(p–1)α p

∗

|ωε |p dx ∩|x|≥δ0



p –N

|x|α 1 + |x| p–1 dx

δ

0 |x|≤ p–1

p∗

  Q(x) – Q(0)|ωε |p∗ dx

N

+ 2QM CNp ε p

ε p



p –N

dx ε + |x| p–1

∩|x|≥δ0

N

(p–1)α  =O ε p +O εp . Since N ≥ 2p, 1 < α < 

N , p–1



p∗  |Q(x) – Q(0)||ωε |

dx = o(ε

p–1 p

), which implies

∗

Q(x)|ωε |p dx 



∗



|ωε |p dx +

= Qm 





 ∗ Q(x) – Q(0) |ωε |p dx



p–1  ∗ |ωε |p dx + o ε p .

= Qm 

Similarly, we can evaluate the rest of formulas and omit the details here.

Li and Xia Boundary Value Problems (2018) 2018:66

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(iii) supt≥0 Jλ (tωε ) < c∗ . Combining (3.3) with Lemma 2.3, one has  







p–1  |∇ωε |p dx ≤ M1 1 – M1–1 K1 ε) + o ε p ,



p–1  ∗ |ωε |p dx = M2 1 – M2–1 K2 ε) + o ε p ,

  ∗ where M1 = 12 RN |∇uε |p dx, M2 = 12 RN |uε |p dx. Then, using (3.2), (3.3), Lemma 2.3 and Lemma 3.1, we see that 

N

Sp

sup Jλ (tωε ) ≤ t≥0

N–p



p–1  N –1 N – p –1 p M2 K2 (ε) – M1 K1 (ε) + o ε 1+ p p

2N(λQm ) p (p–1)N 

N–p + O ε p +p–N+ pr .

Next, we claim that lim ε

p–1 p

ε→0



 N N – p –1 M2 K2 (ε) – M1–1 K1 (ε) < 0 p p

(3.4)

for ε > 0 small enough, which implies (3.1) holds. According to limε→0 ε p–1 p

K1 K2

N–p M1 . N M2

p–1 p

K1 (ε) = K1 ,

limε→0 ε K2 (ε) = K2 , we know that (3.4) is equivalent to > From the expressions of K1 , K2 , M1 , M2 and uε , a series of computations yield p 3p–2 p N–p p  1 –N K1 2 H(0)CNp ( p–1 ) RN–1 (1 + |x | p–1 ) |x | p–1 dx = p p∗  1 K2 H(0)CNp RN–1 (1 + |x | p–1 )–N |x |2 dx 2

p–p∗ = CNp



N –p p–1

p  ∞

2Np+3p–2N–2

2 –N p 0 (1 + r ) r 2Np+p–2N–2 ∞ 2 –N p 0 (1 + r ) r

 N – p M1 N – p RN |∇uε |p dx  = p∗ N M2 N RN |uε | dx

dr

,

dr

2Np+p–2N

∞ dr N – p p–p∗ N – p p 0 (1 + r2 )–N r p . CNp = 2Np–p–2N  ∞ N p–1 2 )–N r p (1 + r dr 0

Integrating by parts, we have  0

∞

β –1 rβ dr = (1 + r2 )n 2n – β – 1

 0

∞

rβ–2 dr (1 + r2 )n

Then

p K1 (p – 1)(N + 1) p–p∗ N – p , = CNp K2 p–1 N – 2p + 1

p N – p M1 p–p∗ N – p = CNp (p – 1). N M2 p–1

for 2 ≤ β < 2n – 1.

Li and Xia Boundary Value Problems (2018) 2018:66

This implies

K1 K2

>

N–p M1 . N M2

Page 13 of 17

Thus

N

Sp

sup Jλ (tωε ) < t≥0

2N(λQm )

N–p p

= c∗ .



The proof of Lemma 3.2 is complete. Proof of Theorem 1.1 Applying Lemma 2.1 and Lemma 3.2, we obtain

 c = inf max Jλ h(t) ≤ sup Jλ (tωε ) < c∗ . h∈ t∈[0,1]

t≥0

From Lemma 2.2 and the mountain pass theorem, we know that there exists at least one nontrivial solution to Problem (1.5). Since Jλ (u) ≥ Jλ (|u|), Problem (1.5) has at least one nonnegative nontrivial solution. The proof of Theorem 1.1 is complete.  Proof of Theorem 1.2 Consider the following function: h(t) = Jλ (tu)    tp

t p∗ ∗ = |∇u|p + λV (x)|u|p dx – ∗ λ Q(x)|u|p dx p  p  q  t – λ P(x)|u|q dx, t > 0. q  Since V ∈ L1 (), we find that  p    t p∗ tq t ∗ sup h(t) = sup λV (x)|A|p dx – ∗ λ Q(x)|A|p dx – λ P(x)|A|q dx p q  t≥0 t≥0 p     Np  λ  V (x) dx for u = A. ≤ N ( Q(x) dx) N–p N 

Then supt≥0 Jλ (tA) < c∗ for λ < Similarly,

N–p  S(  Q(x) dx) N N–p  QMN  V (x) dx

.

Jλ (tA) = sup h(t) t≥0

 p    t p∗ tq t p p∗ q = sup λV (x)|A| dx – ∗ λ Q(x)|A| dx – λ P(x)|A| dx p q  t≥0 p   q

 q – p (  V (x) dx) q–p < c∗ ≤λ p  pq ( P(x) dx) q–p 

2

for

pq p λ < ( q–p )N

p  (  P(x) dx) N(q–p) pq p N–p  N(q–p) N N QMN (  V (x) dx)

S

.

Li and Xia Boundary Value Problems (2018) 2018:66

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Set N–p  

p S(  Q(x) dx) N S pq N λ∗ = max , N–p  N–p p q–p QMN  V (x) dx N N QMN

p2  (  P(x) dx) N(q–p) , pq  (  V (x) dx) N(q–p)

then we have supt≥0 Jλ (tA) < c∗ for 0 < λ < λ∗ . Similar to the proof of Theorem 1.1, Problem (1.5) has at least one nonnegative nontrivial solution. The proof of Theorem 1.2 is complete.  Proof of Theorem 1.3 Define  K=

1,p

p B(x0 ,δ) |∇u| dx

inf

u∈W0 (B(x0 ,δ))\{0}

(



p

q q B(x0 ,δ) |u| dx)

.

Since p < q < p∗ , as is well known, there exists a function w ∈ W0 (B(x0 , δ)) such that 1,p



p B(x ,δ) |∇w| dx K=  0 p . ( B(x0 ,δ) |w|q dx) q

Thus,  p  

 tq t p p q |∇w| + λV (x)|w| dx – λ P(x)|w| dx sup Jλ (tw) ≤ sup q B(x0 ,δ) t≥0 t≥0 p B(x0 ,δ) q  p q – p ( B(x0 ,δ) |∇w| dx) q–p ≤ p p pq q–p  Pm ( B(x0 ,δ) λ|w|q dx) q–p q

q–p = pq q

∗

Let λ = (

K q–p

N–p p

N(q–p)K q–p QM p N q–p pqS p Pm

p

p

.

λ q–p Pmq–p p(q–p)

) Np+pq–Nq , where Pm = minx∈B(x0 ,δ) P(x), then supt≥0 Jλ (tw) < c∗ for

λ > λ∗ . Similar to the proof of Theorem 1.1, Problem (1.5) has at least one nonnegative  nontrivial solution for λ > λ∗ . The proof of Theorem 1.3 is complete. Proof of Theorem 1.4 Fix n ∈ N , let ϕ1 , ϕ2 , . . . , ϕn ∈ C0∞ (RN ) be smooth functions such that supp ϕj ⊂ B(x0 , δ), j = 1, 2, . . . , n, supp ϕi ∩ supp ϕj = ∅, i = j. We define En = Span{ϕ1 , ϕ2 , . . . , ϕn },  is the set of all symmetric and closed subsets of 1,p WV (), γ (A) is the Krasnoselski genus,

 i(A) = min γ h(A) ∩ ∂Bβλ ), h∈

A ∈ , 1,p

1,p

where  is the set of all odd homomorphisms C 1 (WV (), WV ()). Set cj = inf sup Jλ (u), i(A)≥j u∈A

j = 1, 2, . . . , n.

Li and Xia Boundary Value Problems (2018) 2018:66

Page 15 of 17

Since i(En ) = dim En = n and Jλ (u) ≥ βλ for uλV = ρλ in Lemma 2.1, we find that βλ ≤ c1 ≤ c2 ≤ · · · ≤ cn ≤ sup Jλ (u). u∈En

We now estimate supu∈En Jλ (u). If u ∈ En , one has u = erties of ϕj , we obtain

sup Jλ (u) = sup u∈En

u∈En

≤ sup

 n 

n

j=1 τj ϕj

for τj ∈ R. From the prop-

 Jλ (τj ϕj )

j=1 n  p   τj

p

u∈En j=1

q–p  ≤ pq j=1 n

( λ

q

τj

 |∇ϕj |p + λV (x)|ϕj |p dx – λ q B(x0 ,δ)

p q–p





 P(x)|ϕj |q dx B(x0 ,δ)

q

p q–p B(x0 ,δ) |∇ϕj | dx) p q–p

Pm (



B(x0 ,δ) |ϕj

p

.

|q dx) q–p

Consequently, there exists a n > 0 such that supu∈En Jλ (u) < c∗ for λ > n . Similar to the proof of Theorem 1.1, Problem (1.5) has at least n pairs of nonnegative nontrivial solutions. The proof of Theorem 1.4 is complete. 

4 Conclusion In this paper, we study the following quasilinear Neumann problem with critical Sobolev exponent: ⎧ ⎨–εp  u + V (x)|u|p–2 u = Q(x)|u|p∗ –2 u + P(x)|u|q–2 u, p ⎩|∇u|p–2 ∂u = 0, ∂ν

x ∈ , x ∈ ∂,

where the weight functions V (x) is continuous in  and Q(x), P(x) are continuous on . ∗ Due to the lack of compactness of the embedding of W 1,p () → Lp () and the fact that the weight function V (x) may be unbounded close to the boundary ∂, some classical methods may not directly be applied to our problem. We introduce a suitable weighted Sobolev space and add restrictions on the weight functions Q(x) and P(x) to prove the corresponding functional of problem satisfies (PS)c -condition in a suitable range by the Lions concentration-compactness principle, then apply the mountain pass lemma, the existence and multiplicity of nontrivial solutions are obtained.

Acknowledgements The authors would like to thank the editor and the referees for their valuable comments and suggestions, which improved the quality of our manuscript. Funding This work is supported by the National Natural Science Foundation of China (11601122) and by DR Fund of Henan University of Technology (150152). Abbreviations Not applicable Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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